# Conditioning over Conditional probability? (also: $\phi$-mixing sequences)

For two sub $\sigma-$fields $\mathscr{F}$ and $\mathscr{G}$ of a probability space $(\Omega , \mathscr{A} , P)$ we define $\phi$ mixing as follows: $$\phi(\mathscr{F},\mathscr{G}) = \sup \{ |P(G|F) - P(G)| : F \in \mathscr{F} , G \in \mathscr{G}, P(F)>0 \}$$

Now my question is how does one go about conditioning $\phi(\mathscr{F},\mathscr{G})$ with respect to another $\sigma-$field $\mathscr{H}$, that is, $\phi(\mathscr{F},\mathscr{G}| \mathscr{H})$.

Any help will be appreciated.

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I'm not sure to understand your question. You want to extend the definition of the ϕ-coefficient to a notion of conditional ϕ-coefficient ? –  Stéphane Laurent Jun 11 '12 at 17:37
Yes. I want to know what would be a natural way of conditioning the $\phi$-coefficient. Or at least how one should go about thinking about it. Thanks in advance! –  Rohit Jun 11 '12 at 22:20
Have you tried to type "conditional mixing" in Google ? Maybe you could be inspired by this notion of conditional mixing : ism.ac.jp/editsec/aism/pdf/061_2_0441.pdf –  Stéphane Laurent Jun 16 '12 at 18:31